The relativistic representation at once provides a visual explanation for c as a limiting velocity (given the invariance of world-lines described above, a vector along the x-axis will have the maximum possible relative extension in space, equal to the observer's extension in time) and an explanation for the invariant measure of c (again by the invariance of world-lines, an observer will always measure light as moving as far along the space axis as she moves along the time axis). These aspects of light have remained inexplicable by adherence to the Minkowski diagram, and have prevented the realization of Minkowski's original vision -- that "physical laws might find their most perfect expression" in a two-dimensional spacetime projection.
The relationships projected in the relativistic diagram can be expressed in terms of three corollaries thus far:
1. The speed of light is a limit. If the world-lines of bodies in relative motion are taken as having the same spacetime interval but with varying spatial and temporal components according to their relative spacetime trajectories, the limiting spatial velocity is the interval of a world-line along the space axis measured in terms of the same interval along the corresponding time axis. (A vector drawn along the x-axis in figure 4 to represent a ray of light extends as far along the x-axis as time elapses for the observer in the duration of the diagram. There is no vector that can extend further (i.e., move faster) in space than one that has a temporal component of zero.)
2. The speed of light is invariant. Due to the equivalence of the observer's and the observed world-lines, each observer will measure light as traveling the same distance in space as time elapses in that observer's reference frame, and though the measure of the spatial distance traveled by a beam of light between events will vary between reference frames, the rate will always be agreed upon.
3. The speed of light and the speed of time are equivalent. Given the equivalence of world-lines, given the perpendicular relationship between space and time expressed by the Lorentz transformations, and given the world-line of light as lying along the x-axis, distance in time must be equal to distance in space: one second in time is the same distance, but in a perpendicular direction, as ~300,000 km in space.
Time as motion in space
As the relativistic diagram shows, and as the Lorentz transformations express, each body has its own clock and moves in time with its own orientation in space; to move in time in one coordinate system is to move partly in time, partly in space according to another; the time of a body considered to be in motion relative to another moves across the space of the other.
It is difficult, perhaps impossible to conceive how our everyday experience could involve "moving", space-wise, in time -- especially when our experience is of "maintaining" or "enduring" (rather than "moving") more-or-less parallel in time with everything we commonly observe. But the difficulty is to be expected, considering that temporal motion is in a fourth dimension, and we are creatures attuned to a three-dimensional experience distinct from our sense of duration. It is even more incomprehensible that to move in time is to move 300,000 km across space -- and in just one second. We don't seem to be moving at any great speed even relative to the stars; how could the whole universe of mass be moving at c in time although even on a galactic scale we seem to be moving as-if in a common-sense, three-dimensional order. It is a question ripe for speculation, but the phenomenon itself has immediate, informative implications.
Time and Momentum
The notion of a spacetime continuum, and of time as moving in space, indicates a dynamic aspect of time than is not fully appreciated even in relativity theory, due in part perhaps to a residue of the pre-relativistic and common-sense regard for space and time as being independent and fundamentally different. But if spacetime is a continuum, if time moves in space, or rather, if a body moves in space by moving in time, then temporal motion must be dynamic, and possessing of momentum and kinetic energy.
The component of relative time in momentum is obscured even in the relativistic formulation p = m0v / SQRT(1-v2/c2).
If for the sake of isolating considerations of space and time in momentum we set m0to unity, and for simplicity, as before, set v proportional to c, we have
pst = v / SQRT(1-v2)
Projecting this on a spacetime diagram in figure 5 with body B impinging on A, we set t to unity, x = vt, and t' = SQRT(1-x2) and thereby arrive at
pst = x/t'
(Note: Having established that s, the "invariant interval" is just the proper time of an observed body, and given the invariance of the length of world-lines, we can henceforth dispense with s, and return to using t' as the relative time of the observed, as in figure 2.)
Next Page 1 | 2 | 3 | 4 | 5 | 6
(Note: You can view every article as one long page if you sign up as an Advocate Member, or higher).
1
